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The only way a function can be both even and odd is for it to ignore the input, i.e. for it to be a constant function. e.g. f(x)=4 is both even and odd.

An even function is one where f(x)=f(-x), and an odd one is where -f(x)=f(-x).

This doesn't make sense. Let's analyze.

For a function to be even, f(-x)=f(x). For a function to be odd, f(-x)=-f(x).

In this case, f(x)=4, and f(-x)=4. As such, for the first part of the even-odd definition, we have 4=4, which is true, making the function even. However, for the second part of it, we have 4=-4 (f(-x)=4, but -f(x)=-4), which is not true. Therefore constant functions are even because f(-x)=f(x), but not odd because f(-x)!=-f(x).

Q: How a function is even and odd?

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Yes. Along with the tangent function, sine is an odd function. Cosine, however, is an even function.

Yes f(x)=0 is both even and odd

It is an odd function. Even functions use the y-axis like a mirror, and odd functions have half-circle rotational symmetry.

yes

If you know that a function is even (or odd), it may simplify the analysis of the function, for several purposes. One example is the calculation of definite integrals: for an odd function, the integral of a function from (-x) to (x) (note 1) is zero; for an even function, this integral is twice the integral of the function from (0) to (x). Note 1: That is, the area under the function; for negative values, this "area" is taken as negative) is

Related questions

An even function is a function that creates symmetry across the y-axis. An odd function is a function that creates origin symmetry.

I find it convenient to express other trigonometric functions in terms of sine and cosine - that tends to simplify things. The secant function is even because it is the reciprocal of the cosine function, which is even. The tangent function is the sine divided by the cosine - an odd function divided by an even function. Therefore it is odd. The cosecant is the reciprocal of an odd function, so it is naturally also an odd function.

An even function is symmetric about the y-axis. An odd function is anti-symmetric.

both

Yes. Along with the tangent function, sine is an odd function. Cosine, however, is an even function.

Yes f(x)=0 is both even and odd

An even number can be divided by 2 evenly. An odd number will have a remainder of 1 when divided by 2. A function can be either.

For an even function, f(-x) = f(x) for all x. For an odd function, f(-x) = -f(x) for all x.

An even function is symmetric about the y-axis. If a function is symmetric about the origin, it is odd.

You can tell if a function is even or odd by looking at its graph. If a function has rotational symmetry about the origin (meaning it can be rotated 180 degrees about the origin and remain the same function) it is an odd function. f(-x)=-f(x) An example of an odd function is the parent sine function: y=sinx If a function has symmetry about the y-axis (meaning it can be reflected across the y-axis to produce the same image) it is an even function. f(x)=f(-x) An example of an even function is the parent quadratic function: y=x2

You can tell if a function is even or odd by looking at its graph. If a function has rotational symmetry about the origin (meaning it can be rotated 180 degrees about the origin and remain the same function) it is an odd function. f(-x)=-f(x) An example of an odd function is the parent sine function: y=sinx If a function has symmetry about the y-axis (meaning it can be reflected across the y-axis to produce the same image) it is an even function. f(x)=f(-x) An example of an even function is the parent quadratic function: y=x2

It is an odd function. Even functions use the y-axis like a mirror, and odd functions have half-circle rotational symmetry.